Singular integral inequalities and stability of semilinear parabolic equations. (English) Zbl 0915.34057

Medveď, Milan

Arch. Math., Brno 34, No. 1, 183-190 (1998).

The abstract Cauchy initial value problem given for \[ \tfrac {du}{dt} + Au = f(t,u), \quad u \in X, \qquad u(0)=u_0 \in X, \] is considered, where \(X\) is an appropriate Banach space, \(A:X \to X\) is a sectorial operator and \(f\) has some more specific properties. Conditions are presented for the existence of a solution \(u\) defined on \([0,+\infty)\) such that \(\lim _{t\to \infty }\| u(t)\| =0\).

Reviewer: Š.Schwabik (Praha)

MSC:

34G20	Nonlinear differential equations in abstract spaces
34D05	Asymptotic properties of solutions to ordinary differential equations
35K55	Nonlinear parabolic equations
35B35	Stability in context of PDEs
34G10	Linear differential equations in abstract spaces

Keywords:

integral inequality; parabolic equation; stability